Magnetic Resonance Imaging

Shi Chen & Pan Hui

Chapter 13

Outline

• We first explore the instrumentation necessary to create MR images.

• Then we present the image formation process.– Imaging equations– Computer algorithms

• Finally, we discuss the factors affecting image quality.

Instrumentation

• MRI System components

http://www.fas.org/irp/imint/docs/rst/Intro/Part2_26c.html

1 the main magnet2 a set of coils3 resonators4 electronics5 a console

• The magnet, gradient coils, and RF coils must be isolated from the electronic noise of the outside world in order to prevent interfering signals.

• Faraday Cage– All electronic signals go through this filters to ensure

that no noise is present.

Magnet

• Cylindrical superconducting magnet– The most common type used in MRI systems

Main magnet with the patient table

The console for operating the scannerhttp://www.ahtiny.com/equipments/Imaging/MRI_Scanner.htm

superconducting magnets

• There are two major challenges in the design and maintenance of superconducting magnets.– The homogeneity of the magnetic field within the bore

must be maintained at better than +5ppm.

– The minimization of the so-called fringe field—the magnetic field that is outside the bore of the magnet.

Gradient Coils

• Definition– The gradient coils fit just inside the bore of the

magnet.

• Function– To provide a temporary change in the magnitude B0 of

the main magnetic field as a function of position in the magnet bore.

• There are usually 3 orthogonal gradient coils.

Gradient coils provide the means to choose slices of the body for selective imaging. In this way, it can image slices.

Gradient Coils

• If all three coils are turned on at the same time with strengths the main field is given by

• is often written in vector form as

• B can be written using a dot product notation as

𝑩=(𝐵0+𝐺𝑥 𝑥+𝐺𝑦 𝑦+𝐺𝑧 𝑧 ) �̂�

𝑮=(𝐺𝑥 ,𝐺𝑦 ,𝐺𝑧)

𝑩=(𝐵0+𝑮 ∙𝒓 ) �̂�

Radio-Frequency Coils

• RF Coils serve to both induce spin precession and to have currents induced in them by the spin system.

• There are two types of RF coils:– Volume coils– Surface coils

Radio-Frequency Coils

(a)saddle coil (b)birdcage coil (c)surface coil

There are many other volume coils, such as knee coils, neck coils, etc.

MRI Data Acquisition

• Encoding Spatial Position– The +z-direction is from the head to the feet; – +y is oriented posterior(back) to anterior(front); – +x is oriented right to left.

• In this scenario,– We could get a axial image by holding z constant;– We could get a coronal image by holding y constant;– We could get a sagittal image by holding x constant;

Laboratory coordinates in an MR scanner

Frequency encoding

• Larmor frequency

• Where the dependence of Larmor frequency v(r) on spatial position r=(x,y,z) is made explicit.

𝑣 (𝒓 )=γ (𝐵0+𝑮 ∙𝒓 )

Slice selection

• Principle of Slice selection

• When G has only one nonzero component z

𝑣 (𝑧 )=γ (𝐵0+𝐺𝑧 𝑧)

G=(0,0,)

Effect on the main magnetic field from a z-gradient

• There are 3 parameters to select slices:• z-gradient strength Gz, • RF center frequency,

• And RF frequency range,

𝑣=𝑣1 +𝑣2

2

△𝑣=¿ 𝑣2−𝑣1∨¿

• We find that the v1 and v2 yield the slice boundaries,

• Where =v() and =v()𝑧1=

𝑣1−γ 𝐵0

γ𝐺𝑧𝑧 2=

𝑣2−γ 𝐵0

γ𝐺𝑧

• Slice position is therefore given by

• Slice thickness is given by

𝑧=𝑧1+𝑧 2

2=𝑣−𝑣0

γ𝐺𝑧

△𝑧=|𝑧2−𝑧 1|=△𝑣γ𝐺𝑧

)()(vvvArectvS

We know that slice selection uses a constant gradient together with an RF excitation over a range of frequencies[v1,v2]. We can desire a signal whose frequency content is:

According to Fourier transform theory, the signal itself should be:

tvjevtcvAts 2)(sin)(

1. The gradient is constant during RF excitation.2. The RF excitation is short.

If RF signal B1(t) = s(t) has on the spin system, the final tip angle after an RF excitation pulse of duration tp and is repeated here:

pt e dttB0 1 )(

Where is the envelope of the RF excitation evaluated in the rotating coordinate system.

)(1 tBe

For isochromats whose Larmor frequency is v, the excitation signal in the rotating coordinate system is:

vtje etstB 21 )()(

a slice selection waveform

envelop of this slice selection

Refocusing Gradients

During RF excitation, the spin system within the excited slab is undergoing forced precession. The slice profile reveals differences in the final tip angels and hence implies different transverse magnetizations experienced at different z positions.

The effect of slice dephasing:During forced precession, the spins at the “lower” edge of the slice are processing slower than those at the “higher” edge.

Why?

Because system use different Larmor frequencies. As a result of this, the spins become out of phase with each other across the slice.

Refocusing Gradients

A Simple Pulse Sequence

After the RF waveform is completed., another gradient is applied to refocus the spins within the slice. After this, we expect to find an FID arising from the excited spins in the slice that was selected.

At the completion of the refocusing gradient pulse, the phase angle of all magnetization vectors in the same, and the signal from these magnetization vectors will add constructively.

If no dephasing were present across the selected slice, then we would expect the FID to begin at the center of the RF pulse.

Because of dephasing, the appearance of the FID is delayed until near the conclusion of the refocusing lobe.

Assuming the slice is fairly thin so there is no z variation. There will be a spatial variation of transverse magnetization immediately after RF excitation, which is M xy(x,y;0+). So the received signal can be written as:

dxdyeeyxMAts yxTttvjxy

),(/2 20)0;,()(

dxdyeyxAMe yxTtxy

tvj ),(/2 20 )0;,(

gl

Some details must make clear:

1. The FID decays more rapidly than T2; therefore, we must view either as an idealized signal model, or one that applies only for very short time intervals, where the difference in decay rates is negligible.

2. It should be noted that t = 0 represents the center of the slice selection RF waveform.

3. The equation ignores the short time tp it takes for the FID to actually appear after the refocusing lobe of the slice select gradient.

),(/ 2)0;,(),( yxTteyxAMyxf

For clarity, define the effective spin density as:

Which represents the MR quantity that is being imaged here.

dxdyyxfets tvj

),()( 02

The received signal is always demodulated in MRI hardware, yielding the baseband signal:

dxdyyxftsets tvj

),()()( 020

Readout Gradient

The first concept required for spatially encoding MR signals is frequency encoding. In frequency encoding, a gradient is turned on during the FID, causing the Larmor frequencies to be spatially dependent.

Readout Gradient

The direction of the frequency encoding gradient is called the readout direction because the signal that is “read out” is spatially encoded in that direction.

The Larmor frequencies during a frequency encode gradient are given by:

)()( 0 xGBxv x

Using Larmor frequency in received signal equation:

dxdyeeyxMAts yxTttxGvjxy

x ),(/)(2 20)0;,()(

dxdyeeyxAMe xtGjyxTtxy

tvj x 2),(/2 20 )0;,(

Using the definition of effective spin density, above equation can covert to:

dxdyeyxfts xGj x2),()(

The spatial frequency variable in the x-direction as tGu x

Which has units of inverse length.The spatial frequency variable in the y direction is :

0vDenoting F(u,v) as the 2-D Fourier transform of f(x,y), we can now make the identity:

)()0,( 0xGusuF

Which shows that the demodulated FID represents a certain “scan” of the 2-D Fourier space of the effective spin density.

In magnetic resonance imaging, Fourier space is usually referred to as k-space. The k-space variables can be identified with our Fourier frequencies,

vkuk

y

x

Polar Scanning

A more general gradient involving both an x- and a y-component can be used to encode the Larmor frequency:

)(),( 0 yGxGByxv yx A baseband signal given by :

dxdyeyxfts tyGxGj yx

)(20 ),()(

Polar Scanning

A pulse sequence for arbitrary polar scan

Polar Scanning

The Fourier frequencies can be defined as:

tGvtGu

y

x

The implied Fourier trajectory is a ray emanating from the origin in the direction:

x

y

GG1tan

Polar Scanning

A Fourier trajectory for a polar scan.

Gradient Echoes

A new mechanism to create an echo,: gradient echo.This idea can be readily connected to both the Fourier trajectories and the intuitive idea of spins realigning themselves.